we did this as a class project before. we dropped the egg from a 3rd story building. our group used bubble wrap, cardboard, and foam. we cutted the foam so it would fit the eggs perfect(we had two raw eggs), wraped the eggs in bubble wrap and placed the eggs in the foam and then placed everything in the cardboard box. i seen other groups in my class do some cool stuff. like i seen a group used a rocket made out of cardboard, and used the tip of the rocket as protector. parachute works, but if you want it to land in a given area(like we had to make our eggs land in a specific area), it goes flying everywhere.

Take the biggest feather pillow you can find and secure the egg on top of the feathers in the case. Bubble wrap should work for extra cushioning. The downside of course will be if the pillows rolls over in free fall putting the egg on the ground before the feather cushioning.